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OMNITRUNCATION

Omnitruncation

Geometric operation

In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope

Omnitruncation

Conway polyhedron notation

Method of describing higher-order polyhedra

edges. Meta (in its non-indexed form) is also called cantitruncation or omnitruncation. Note that 0 here does not mean the same as for augmentation operations:

Conway polyhedron notation

Conway_polyhedron_notation

Uniform tiling

Vertex-transitive tiling of the plane by regular polygons

until edges disappear), and cantellation (cutting edges and vertices). Omnitruncation is an operation that combines truncation and cantellation. Snubbing

Uniform tiling

Uniform_tiling

Expansion (geometry)

Geometric operation on convex polytopes

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Expansion (geometry)

Expansion_(geometry)

Uniform 8-polytope

Polytope contained by 7-polytope facets

shown below, 4 single-ringed, 3 truncations (2 rings), and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing

Uniform 8-polytope

Uniform_8-polytope

Uniform polytope

Isogonal polytope with uniform facets

are applied at once, the operation can be more generally called an omnitruncation. One special operation, called alternation, removes alternate vertices

Uniform polytope

Uniform_polytope

List of polygons, polyhedra and polytopes

Truncation (geometry) Bitruncation Cantellation Runcination Sterication Omnitruncation Expansion (geometry) Snub (geometry) Alternation (geometry) Dual polyhedron

List of polygons, polyhedra and polytopes

List_of_polygons,_polyhedra_and_polytopes

Truncation (geometry)

Operation that cuts polytope vertices, creating a new facet in place of each vertex

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Truncation (geometry)

Truncation_(geometry)

Barycentric subdivision

Method for dividing a simplicial complex

can have faces that are not simplices. This is the dual operation to omnitruncation. The vertices of the barycentric subdivision correspond to the faces

Barycentric subdivision

Barycentric_subdivision

Omnitruncated polyhedron

symbols are listed first, followed by the actual mixed Wythoff symbols. Omnitruncations are also called cantitruncations or truncated rectifications (tr),

Omnitruncated polyhedron

Omnitruncated_polyhedron

List of Greek and Latin roots in English/P–Z

entrench, entrenchment, intrench, intrenchment, obtruncate, obtruncation, omnitruncation, retrench, retrenchment, tranche, tranchet, trench, trenchancy, trenchant

List of Greek and Latin roots in English/P–Z

List_of_Greek_and_Latin_roots_in_English/P–Z

Uniform 4-polytope

Class of 4-dimensional polytopes

14 Bicantitruncation t1,2,3{p,q,r} Cantitruncated dual {r,q,p}. 15 Omnitruncation (runcicantitruncation) t0,1,2,3{p,q,r} Application of all three operators

Uniform 4-polytope

Uniform_4-polytope

Alternation (geometry)

Removal of alternate vertices

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Alternation (geometry)

Alternation_(geometry)

Bitruncation

Operation in Euclidean geometry

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Bitruncation

Heptellated 8-simplexes

family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex;

Heptellated 8-simplexes

Heptellated_8-simplexes

Uniform 5-polytope

Five-dimensional geometric shape

q,2r,s} Alternated truncated rectification ht0,1,2,3{p,q,r,s} Alternated runcicantitruncation Full snub ht0,1,2,3,4{p,q,r,s} Alternated omnitruncation

Uniform 5-polytope

Uniform_5-polytope

Pentellated 6-simplexes

Uniform 6-polytope

regular 6-simplex. Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes) Omnitruncated heptapeton Great terated tetradecapeton

Pentellated 6-simplexes

Pentellated_6-simplexes

Stericated 5-cubes

1+3{\sqrt {2}}\right)} Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson) Omnitruncated penteract Omnitruncated triacontaditeron

Stericated 5-cubes

Stericated_5-cubes

Hexicated 7-simplexes

Type of 7-polytope

hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active. The omnitruncated

Hexicated 7-simplexes

Hexicated_7-simplexes

List of Greek and Latin roots in English/T

entrench, entrenchment, intrench, intrenchment, obtruncate, obtruncation, omnitruncation, retrench, retrenchment, tranche, tranchet, trench, trenchancy, trenchant

List of Greek and Latin roots in English/T

List_of_Greek_and_Latin_roots_in_English/T

Point groups in four dimensions

diagram = 4-polytope net Omnitruncation omni. 5-cell omni. demitesseract omni. tesseract omni. 24-cell omni. 120-cell Omnitruncation dual net Coxeter diagram

Point groups in four dimensions

Point_groups_in_four_dimensions

Stericated 5-simplexes

duoprisms). Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson) Omnitruncated hexateron Great cellated dodecateron

Stericated 5-simplexes

Stericated_5-simplexes

Truncated order-6 hexagonal tiling

tiling, h2{4,6} By *663 symmetry, this tiling can be constructed as an omnitruncation, t{(6,6,3)}: The dual to this tiling represent the fundamental domains

Truncated order-6 hexagonal tiling

Truncated_order-6_hexagonal_tiling

Snub polyhedron

Polyhedron resulting from the snub operation

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Snub polyhedron

Snub_polyhedron

Snub (geometry)

Geometric operation applied to a polyhedron

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Snub (geometry)

Snub_(geometry)

Rectification (geometry)

Operation in Euclidean geometry

operators v t e Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations t0{p,q} {p,q} t01{p,q} t{p,q} t1{p,q} r{p,q} t12{p,q} 2t{p

Rectification (geometry)

Rectification_(geometry)

Truncated 24-cells

16-cell, with two families of truncated octahedral cells. D4 [31,1,1]: An omnitruncation of the demitesseract, with three families of truncated octahedral cells

Truncated 24-cells

Truncated_24-cells

Zonohedron

Convex polyhedron projected from hypercube

of regular-faced zonohedra, there are three Archimedean solids, all omnitruncations of the regular forms: The truncated octahedron, with 6 square and 8

Zonohedron

Truncated order-6 pentagonal tiling

With mirrors An alternate construction exists from the [(5,5,3)] family, as the omnitruncation t012(5,5,3). It is shown with two (colors) of decagons.

Truncated order-6 pentagonal tiling